6 - 1 © 1998 Prentice-Hall, Inc. Chapter 6 Sampling Distributions

6 - 6 - 11

© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.

Chapter 6Chapter 6

Sampling DistributionsSampling Distributions

6 - 6 - 22


Learning ObjectivesLearning Objectives

1.1. Describe the properties of estimatorsDescribe the properties of estimators

2.2. Explain sampling distributionExplain sampling distribution

3.3. Describe the relationship between Describe the relationship between populations & sampling distributionspopulations & sampling distributions

4.4. State the Central Limit TheoremState the Central Limit Theorem

5.5. Solve a probability problem involving Solve a probability problem involving sampling distributionssampling distributions

6 - 6 - 33


Inferential StatisticsInferential Statistics

6 - 6 - 44


Types of Types of Statistical Statistical

ApplicationsApplications

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

6 - 6 - 55



1.1. InvolvesInvolves EstimationEstimation Hypothesis Hypothesis

testingtesting

6 - 6 - 66




testingtesting

Population?Population?

6 - 6 - 77




testingtesting

2.2. PurposePurpose Make decisions Make decisions

about population about population characteristicscharacteristics

Population?Population?

6 - 6 - 88


Inference ProcessInference Process

6 - 6 - 99



PopulationPopulation

6 - 6 - 1010




SampleSample

6 - 6 - 1111




SampleSample

Sample Sample statistic statistic

((XX))

6 - 6 - 1212




SampleSample

Sample Sample statistic statistic

((XX))

Estimate Estimate & test & test populatiopopulation n parameterparameter

6 - 6 - 1313


1.1. Random variables used to estimate a Random variables used to estimate a population parameterpopulation parameter Sample mean, sample proportion, sample Sample mean, sample proportion, sample

medianmedian

2.2. Example: Sample meanExample: Sample meanxx is an estimator is an estimator of population mean of population mean IfIfxx = 3 then 3 is the = 3 then 3 is the estimateestimate of of

3.3. Theoretical basis is sampling distributionTheoretical basis is sampling distribution

EstimatorsEstimators

6 - 6 - 1414


Sampling DistributionsSampling Distributions

6 - 6 - 1515


1.1. TheoreticalTheoretical probability distribution probability distribution

2.2. Random variable is Random variable is sample statisticsample statistic Sample mean, sample proportion etc.Sample mean, sample proportion etc.

3.3. Results from drawing Results from drawing allall possible possible samples of a samples of a fixedfixed size size

4.4. List of all possible [List of all possible [x, x, PP((x) ] pairsx) ] pairs Sampling distribution of meanSampling distribution of mean

Sampling Sampling DistributionDistribution

6 - 6 - 1616


DevelopingDevelopingSampling Sampling

DistributionsDistributions

Population size, Population size, NN = 4 = 4

Random variable, Random variable, xx, , is # televisions ownedis # televisions owned

Values of Values of xx: 1, 2, 3, 4: 1, 2, 3, 4

Equally distributed (p=1/4)Equally distributed (p=1/4)

© 1984-1994 T/Maker Co.

Suppose there’s a Suppose there’s a population ...population ...

6 - 6 - 1717


Population Population CharacteristicsCharacteristics

6 - 6 - 1818



Summary MeasuresSummary Measures

6 - 6 - 1919



Summary MeasuresSummary Measures

12.1)(

5.2)(

1

2

1

)(

ipi

ip

N

ii

N

iii

XX

XX

12.1)(

5.2)(

1

2

1

)(

ipi

ip

N

ii

N

iii

XX

XX

6 - 6 - 2020



.0

.1

.2

.3

1 2 3 4

.0

.1

.2

.3

1 2 3 4

Population DistributionPopulation DistributionSummary MeasuresSummary Measures

12.1

5.2

12.1

5.2

6 - 6 - 2121


Let’s Draw All Let’s Draw All Possible Samples of Possible Samples of

Size Size nn = 2 = 2

6 - 6 - 2222



Size Size nn = 2 = 2

1st 2nd ObservationObs 1 2 3 4

1 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4


1 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4

16 Samples16 Samples

Sample with replacementSample with replacement

6 - 6 - 2323



Size n=2Size n=2


1 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4


1 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4


1 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0


1 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0

16 Samples16 Samples 16 Sample Means16 Sample Means

Sample with replacementSample with replacement

6 - 6 - 2424


Sampling Sampling DistributionDistribution

of All Sample of All Sample MeansMeans

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0

X

P(X)

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0

X

P(X)


1 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0


1 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0

16 Sample Means16 Sample Means Sampling Sampling DistributionDistribution

6 - 6 - 2525


Summary Measures Summary Measures ofof

All Sample Means All Sample Means (n=16)(n=16)

5.216

0.45.10.11

n

Xn

ii

x 5.216

0.45.10.11

n

Xn

ii

x

79.0

16

5.20.45.25.15.20.1 222

1

2

n

Xn

ixi

x

79.0

16

5.20.45.25.15.20.1 222

1

2

n

Xn

ixi

x

6 - 6 - 2626


Comparison of Comparison of Population & Sampling Population & Sampling

DistributionDistribution

6 - 6 - 2727




.0

.1

.2

.3

1 2 3 4

P(X)

.0

.1

.2

.3

1 2 3 4

P(X)


112. 112.

2 5. 2 5.

6 - 6 - 2828




.0

.1

.2

.3

1 1.5 2 2.5 3 3.5 4

X

P(X)

.0

.1

.2

.3

1 1.5 2 2.5 3 3.5 4

X

P(X)

.0

.1

.2

.3

1 2 3 4

P(X)

.0

.1

.2

.3

1 2 3 4

P(X)

PopulationPopulation Sampling DistributionSampling Distribution

x 2 5. x 2 5.

x 0 79. x 0 79. 112. 112.

2 5. 2 5.

6 - 6 - 2929


Standard Error of Standard Error of MeanMean

1.1. Standard deviation of all possible Standard deviation of all possible sample means,sample means,xx Measures scatter in all sample means,Measures scatter in all sample means,xx

2.2. Less than pop. standard deviationLess than pop. standard deviation

6 - 6 - 3030


Standard Error of Standard Error of MeanMean

1.1. Standard deviation of all possible Standard deviation of all possible sample means,sample means,xx Measures scatter in all sample means,Measures scatter in all sample means,xx

2.2. Less than pop. standard deviationLess than pop. standard deviation

3.3. Formula (sampling with replacement)Formula (sampling with replacement)

x n

x n

6 - 6 - 3131


Properties of Sampling Properties of Sampling Distribution of MeanDistribution of Mean

6 - 6 - 3232


Properties of Properties of Sampling Sampling

Distribution of MeanDistribution of Mean1.1. UnbiasednessUnbiasedness

Mean of sampling distribution equals population Mean of sampling distribution equals population meanmean

2.2. EfficiencyEfficiency Sample mean comes closer to population mean Sample mean comes closer to population mean

than any other unbiased estimatorthan any other unbiased estimator

3.3. ConsistencyConsistency As sample size increases, variation of sample As sample size increases, variation of sample

mean from population mean decreasesmean from population mean decreases

6 - 6 - 3333


UnbiasednessUnbiasedness

X

P(X)

CA

UnbiasedUnbiased BiasedBiased

6 - 6 - 3434


EfficiencyEfficiency

X

P(X)

A

B

Sampling Sampling distribution distribution of medianof median

Sampling Sampling distribution distribution

of meanof mean

6 - 6 - 3535


ConsistencyConsistency

X

P(X)

A

B

Smaller Smaller sample sample

sizesize

Larger Larger sample sample

sizesize

6 - 6 - 3636


Sampling from Sampling from Normal PopulationsNormal Populations

6 - 6 - 3737



Central TendencyCentral TendencyCentral TendencyCentral Tendency

6 - 6 - 3838



Central TendencyCentral TendencyCentral TendencyCentral Tendency x x

6 - 6 - 3939



Central TendencyCentral Tendency

DispersionDispersion

Sampling Sampling withwith replacementreplacement




x n

x n

x x

6 - 6 - 4040


= 50

= 10

X = 50

= 10

X








Population DistributionPopulation DistributionPopulation DistributionPopulation Distribution

x n

x n

x x

6 - 6 - 4141


= 50

= 10

X = 50

= 10

X









Sampling DistributionSampling DistributionSampling DistributionSampling Distribution

x n

x n

x x

X = 50- XX = 50- X

n =16n =16

XX = 2.5 = 2.5

n = 4n = 4

XX = 5 = 5

6 - 6 - 4242


Standardizing Standardizing Sampling Distribution Sampling Distribution

of Meanof Mean

Suppose you want to make probability statements about the sampling distribution...

Suppose you want to make probability statements about the sampling distribution...

6 - 6 - 4343



of Meanof Mean

XX

X

XX

X

Sampling DistributionSampling Distribution

6 - 6 - 4444



of Meanof Mean

XX

X

XX

X


= 0

= 1

Z = 0

= 1

Z

Standardized Normal Distribution


6 - 6 - 4545



of Meanof Mean

XX

X

XX

X


= 0

= 1

Z = 0

= 1

Z

ZX X

n

x

x

Z

X X

n

x

x



6 - 6 - 4646


Thinking ChallengeThinking Challenge

You’re an operations You’re an operations analyst for AT&T. Long-analyst for AT&T. Long-distance telephone calls distance telephone calls are normally distribution are normally distribution with with = 8 = 8 min. & min. & = 2 = 2

min. If you select random min. If you select random samples of samples of 25 25 calls, what calls, what percentage of the percentage of the samplesample means means would be between would be between 7.87.8 & & 8.2 8.2 minutes?minutes?

© 1984-1994 T/Maker Co.

AloneAlone GroupGroup Class Class

6 - 6 - 4747


Sampling Sampling Distribution Distribution

Solution*Solution*

8

X = .4

7.8 8.2 X8

X = .4

7.8 8.2 X


ZX

n

ZX

n

7 8 82 25

50

8 2 82 25

50

..

..

ZX

n

ZX

n

7 8 82 25

50

8 2 82 25

50

..

..

0

= 1

-.50 Z.500

= 1

-.50 Z.50

.3830.3830.3830.3830

.1915.1915.1915.1915



6 - 6 - 4848


Sampling from Sampling from Non-Normal Non-Normal PopulationsPopulations

6 - 6 - 4949



Central TendencyCentral TendencyCentral TendencyCentral Tendency

6 - 6 - 5050



Central TendencyCentral TendencyCentral TendencyCentral Tendency x x

6 - 6 - 5151









x n

x n

x x

6 - 6 - 5252


= 50

= 10

X = 50

= 10

X









x n

x n

x x

6 - 6 - 5353


= 50

= 10

X = 50

= 10

X









Sampling DistributionSampling DistributionSampling DistributionSampling Distribution

x n

x n

x x

X = 50- XX = 50- X

n =30n =30

XX = 1.8 = 1.8

n = 4n = 4

XX = 5 = 5

6 - 6 - 5454


Central Limit TheoremCentral Limit Theorem

6 - 6 - 5555


Central Limit Central Limit TheoremTheorem

6 - 6 - 5656


XX


As As sample sample size gets size gets large large enough enough (n (n 30) ...30) ...

6 - 6 - 5757


XX



sampling sampling distribution distribution becomes becomes almost almost normal.normal.

6 - 6 - 5858


XX



sampling sampling distribution distribution becomes becomes almost almost normal.normal.

x n

x n

x x

6 - 6 - 5959


ConclusionConclusion

1.1. Described the properties of estimatorsDescribed the properties of estimators

2.2. Explained sampling distributionExplained sampling distribution

3.3. Described the relationship between Described the relationship between populations & sampling distributionspopulations & sampling distributions

4.4. Stated the Central Limit TheoremStated the Central Limit Theorem

5.5. Solved a probability problem involving Solved a probability problem involving sampling distributionssampling distributions

6 - 6 - 6060


This Class...This Class...

1.1. What was the most important thing you What was the most important thing you learned in class today?learned in class today?

2.2. What do you still have questions about?What do you still have questions about?

3.3. How can today’s class be improved?How can today’s class be improved?

Please take a moment to answer the following questions in writing:

End of Chapter

Any blank slides that follow are blank intentionally.

Documents

6 - 1 © 1998 Prentice-Hall, Inc. Chapter 6 Sampling Distributions